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# An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes

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LIPIcs.ESA.2021.32.pdf
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## Acknowledgements

We would like to thank Arnaud de Mesmay for useful discussions.

## Cite As

Éric Colin de Verdière and Thomas Magnard. An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.32

## Abstract

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. It is known that the problem admits an algorithm with running time f(c)n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n³-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}n^{O(1)}. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Mathematics of computing → Graph algorithms
• Mathematics of computing → Graphs and surfaces
• Mathematics of computing → Topology
##### Keywords
• computational topology
• embedding
• simplicial complex
• graph
• surface
• fixed-parameter tractability

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